2 Probability 1

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MTH-3111 Dr. M. Hassan

Chapter-2

Probability


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Outline

Sample spaces and events.

Interpreting probabilities.

Addition Rules. Conditional probabilities.

Multiplication and total probability rules.

Independence.

Bayes theorem.

Random variables.


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Sample spaces

Random experiment: An experiment that can resultin different outcomes even though it is repeated in thesame manner every time.

Sample space: The set of all possible outcomes of arandom experiment. The sample space is denoted asS.

Importance of sample space:

To model and analyze a random experiment, we must

understand the set of possible outcomes from theexperiment, i.e., an appropriate description of thesample space.


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Sample spaces (Cont.)

A sample space is discrete if it consists of a

finite or countable infinite set of outcomes.

A sample space is continuous if it contains aninterval (either finite or infinite) of real

numbers.

Representation of sample spaces:

(1) Sample space sets.

(2) Tree diagram.


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(1)Sample space sets (Examples)

Coin Toss: S = {H, T}

Roll Single Dice: S = {1,2,3,4,5,6}

1st

quiz score: S = {0, 1, 2,

, 99, 100} Drive time: S = { t : 0 e t e } = [o, ]

State of residence: S = {KL, Selangor, }

Flip 2 coins: S = {HH, HT, TH, TT}

Picking up two parts from a product batch:

S = {gg,gd,dg,dd}


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Sample spaces (Examples) (Cont.)

Select 3 students from class and:(A) classify as male (M) or female (F)

S = {FFF, MFF, FMF, MMF, FFM, MFM, FMM, MMM}

(B) you are only interested in the number of female students

selectedS = {0, 1, 2, 3}

If there were only one defective part in a productbatch, there would be fewer possible outcomes

because ddwould be impossible.S = { gg, gd, dg }

i.e., a sample space is often defined based on theobjectives of the analysis.


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Sampling With replacement

Sampling with replacement:

Items are replaced before the next one is selected..

Possible ordered outcomes (Sampling) are:

S = {aa, ab, ac, bb, ba, bc, cc, ca, cb}

Possible Unordered sampling are:

S = {{a, a}, {a, b}, {a, c}, {b, b}, {b, c}, {c, c}}.

Ordered outcomes are larger than unordered outcomes.

Sometimes the ordered outcomes are needed, but in other cases thesimpler, unordered sample space is sufficient.


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Ordered vs. unordered

Remember:

Permutation: A permutation is an arrangement of

objects in a definite order.

Combination: A combination is a selection of object

with no regard to order.

)!(!

!

rnr

nC

n

r

!

)1)...(2)(1( ! rnnnnPnr


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Sampling Without replacement

Items are not replaced before the next one is selected..Possible ordered outcomes are

S = {ab, ac, ba, bc, ca, cb}

Outcomes in case ofSampling withoutreplacement are smaller than sampling withreplacement.

Sampling without replacement is more commonfor industrial applications.


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(2) Tree diagram

Example 2-4:

An automobile manufacturer provides vehiclesequipped with selected options. Each vehicle is

ordered With or without an automatic transmission,With or without air-conditioning, With one of threechoices of a stereo System, With one of four exteriorcolors. If the sample space consists of the set of all

possible vehicle types, what is the number ofoutcomes in the sample space?

Solution: The sample space contains 48 outcomes.


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Tree diagram (Cont.)

Trees are helpful when there are more thanTrees are helpful when there are more than 22 elements in a subelements in a sub--spacespace


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Events

An event is a subset of the sample space of a

random experiment.

We can be interested in describing new eventsfrom combinations of existing events.


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E

S

Event and sample space


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Events (Cont.)

Example 2.2:

Two connectors are selected and measured. Ifthe objective of the analysis is to consider onlywhether or not the parts conform to themanufacturing specifications, either part mayor may not conform.

The sample space can be represented by thefour outcomes:

S = {yy, yn, ny, nn}


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Events (Cont.)

Example 2.6:

Consider the sample space S = {yy, yn, ny, nn} inExample 2-2. Suppose that the set of all outcomes

for which at least one part conforms is denoted as(Event) E1:

E1 = { yy, yn, ny }

The event E2 in which both parts do not conform,

contains only the single outcome:E2 = {nn}

Example 2.8:


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Venn diagrams

Venn Diagrams:

Graphical means to portray relationships between

sets, and to describe relationships between events.

The random experiment is represented as the points in

the rectangle S. The eventsA (& B,)are the

subsets of points in the indicated regions.

Example: Toss a die and observe the number that appears on the

upper face. Let eventA = Observe an odd number.


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Sets

Because events are subsets (of sample space), we can

use basic set operations such as unions,

intersections, and complements.

A set:

It is a well-defined collection of objects. Each object

in a set is called an element of the set.

Union: The Union of events A and B (A B) (read A or

B) is the event consisting of all outcomes that are

eitherin A orin B orin both events


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B

A

AA BB


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Sets (Cont.)

Intersection:

The Intersection of events A and B, denotedby A B (read A and B), is the eventconsisting of all outcomes that are in both Aand B.

Complement:

the complement of event A, denoted by A, isthe set of all outcomes in the Sample Spacethat are not contained in A.


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B

A

AA BB


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Sets (Cont.)

Mutually exclusive events:

If two events, A and B have no outcomes incommon, they are said to be mutuallyexclusive ordisjoint events. This means thatif one of them occurs, the other cannot.

A B = , [ = the empty set]

Example: Observing odd and even numbers by rolling

single dice.


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B

A

AA BB = =


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Set Rules

A =

A = A

A A' = A A' = S

S' =

' = S (A')' = A


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Set Rules (Cont.)

DeMorgans theorem:

(A B)' = A' B

(A B)' = A' B

Distributive laws:

(A B) C = (A C) (B C)

(A B) C = (A C) (B C)

A B = B %

A B = B %


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BA

BA

(A(A B)B) = AA BB

AA BB (A(A B)B)


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BA

BA

(A B) (A B)

(A(A B) = AB) = A BB

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